Rules of Inference: Which to use for 'If you are...'?

[solomon_13000]

I tried to identify which rules of inference used for:

If you are at least 21 years old, then you can vote.
You are at least 21 years old.
Therefore, you can vote.


My solution:

P -> Q
P
______
.. Q

Is this: Modus Ponens?

 

[pka]

P -> Q
P
______
.. Q
Is this: Modus Ponens?

Yes!

 

[Count Iblis]

I make this statement: "If I'm not mistaken, then Santa exists."

P = "I'm not mistaken"

Q = "Santa exists"


If I'm not mistaken then my statement P --> Q is true. If you apply Modus Ponens then the truth of Q follows. This is assuming that P is true. But that is exactly what my statement claims, so I'm in fact not mistaken and we can thus conclude that Santa exists.

 

[solomon_13000]

If I try hard and I have talent, then I will become a musician.
If I become a musician, then I will be happy.
..If I will not be happy, then I did not try hard or I do not have talent.

P - I try hard
Q - I have talent
R - I will become a musician
S – I will be happy

(P /\ Q) -> R
R -> S
_____________
..~S -> (~P /\ ~Q)


Is this correct? Cause I never used "S" but up to "R".

 

[solomon_13000]

If I try hard and I have talent, then I will become a musician.
If I become a musician, then I will be happy.
..If I will not be happy, then I did not try hard or I do not have talent.

P - I try hard
Q - I have talent
R - I will become a musician
S – I will be happy

(P /\ Q) -> R
R -> S
_____________
..~S -> (~P \/ ~Q)


Is this correct? Cause I never used "S" but up to "R".